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Congruence II Proposition List

    This topic proves the well-known Alternate Interior Angle and Exterior Angle Propositions much as Euclid did for the Euclidean plane. However, Euclid had gaps in these proofs; in fact, his gaps may not be noticed until trying to follow his proofs on a sphere. This topic uses the Crossbar Proposition to close the gap in the Exterior Angle Proposition. Also the SAA Congruence Criterion and the Hypotenuse-Leg Congruence Criterion are detailed. Propositions concerning midpoints, bisectors, and relations between sides and angles of a triangle are proven.

Proposition (Alternate Interior Angle)
    (i) If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines are parrallel.
    (ii) Two lines perpendicular to the same line are parallel.
    (iii) If is any line and is any point not on there exists at least one line through parallel to

Proposition (Exterior Angle Property) An exterior angle of a triangle is greater than either remote interior angle.

Proposition (SAA Congruence Criterion) Given and Then

Proposition (Hypotenuse-Leg Criterion) Two right triangles are congruent if the hypotneuse and a leg of one are congruent repsectively to the hypotenuse and a leg of the other.

Proposition (Midpoints) Every segment has a unique midpoint.

Proposition (Bisectors)
    (i) Every angle has a unqiue bisector
    (ii) Every segment has a unique perpendicular bisector.

Proposition (Larger Angle Larger Side) In a triangle the greater angle lies opposite the greater side and the greater side lies opposite the greater angle.

Proposition (Side Angle Comparison) Given and   if and then if and only if

Proposition (Segment Comparison)
    (i) If and then
    (ii) Given any triangle and point such that then or

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